We present a deterministic comparison-based algorithm that sorts sequences avoiding a fixed permutation $π$ in linear time, even if $π$ is a priori unkown. Moreover, the dependence of the multiplicative constant on the pattern $π$ matches the information-theoretic lower bound. A crucial ingredient is an algorithm for performing efficient multi-way merge based on the Marcus-Tardos theorem. As a direct corollary, we obtain a linear-time algorithm for sorting permutations of bounded twin-width.